# Integration by U-Substitution

## 1. Introduction

When the integrand is formed by a product (or a division, which we can treat like a product) it's recommended the use of the method known as **integration by u-substitution**, that consists in applying the following formula:

## 2. Tips

- Even though it's a simple formula, it has to be applied correctly. Let's see a few tips on how to apply it well:
**Select**A bad choice can complicate the integrand. Supposing we have a product, and one of the factors is monomial (*u*and*dv*correctly:*x*for example). If we consider that^{3}*dv = x*, then by using integration we obtain that We have increased the exponent and this could mean a step back in the process. Something similar happens with fractions (like^{3}*1/x*). If we take*dv = 1/x*, we will obtain*v = log|x|*, and probably end up with a harder integration process.As a rule, we will call

*u*all powers and logarithms; and*dv*exponentials, fractions and trigonometric functions (circular functions).-
**Don't change our minds about the selection:**Sometimes we need to apply the method more than once for the same integral. When this happens, we need to call*u*the result of*du*from the first integral we applied the method to. The same applies to*dv*. If we don't do this, seeing as choosing one option or another involves integration or differentiating, we'll be undoing the previous step and we won't be able to advance. **Cyclic integrals:**Sometimes, after applying integration by u-substitution twice we have to isolate the very integral from the equality we've obtained in order to resolve it.

## 3. Examples

**Example 1** In this integral we don't have an explicit product of functions, but we don't know what the logarithms primitive function is, so we differentiate it, that way

*u = ln(x)*.

**Example 2**It's in our interest to select

*u = x*(to reduce the exponent) but then we're forced that

^{2}*dv = ln(x)*and obtaining

*v*isn't immediate. So we'll select the other case

**More examples:**

**Integration by U-Substitution:**resolved integrals step by step